In
topology, a sub-Stonean space is a
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
Hausdorff space such that any two open
σ-compact disjoint subsets have disjoint compact closures. Related, an F-space, introduced by , is a
completely regular Hausdorff space for which every finitely generated
ideal of the ring of real-valued continuous functions is
principal, or equivalently every real-valued continuous function
can be written as
for some real-valued continuous function
. When dealing with
compact spaces, the two concepts are the same, but in general, the concepts are different. The relationship between the sub-Stonean spaces and F-spaces is studied in Henriksen and Woods, 1989.
Examples
Rickart space In mathematics, a Rickart space (after Charles Earl Rickart), also called a basically disconnected space, is a topological space in which open σ-compact subsets have compact open closures. named them after , who showed that Rickart spaces are rel ...
s and the
corona set
In mathematics, the corona or corona set of a topological space ''X'' is the complement β''X''\''X'' of the space in its Stone–Čech compactification β''X''.
A topological space is said to be σ-compact if it is the union of countably ma ...
s of locally compact σ-compact Hausdorff spaces are sub-Stonean spaces.
See also
*
Extremally disconnected space
*
F-space
References
*
*
*{{Citation , last1=Henriksen, first1=Melvin, last2=Woods, first2=R. G., title=F-Spaces and Substonean Spaces: General Topology as a Tool in Functional Analysis, journal=
Annals of the New York Academy of Sciences, volume=552, issue=1 Papers on General topology and related category theory and topological algebra, pages=60–68, doi=10.1111/j.1749-6632.1989.tb22386.x, issn=1749-6632 , mr=1020774, year=1989
Properties of topological spaces